On the $p$-adic deformation problem for the $K$-theory of semistable schemes

Alberto Merici; Doosung Park; Federico Binda; Tommy Lundemo

arxiv: 2601.14146 · v2 · pith:YAUIFFJ6new · submitted 2026-01-20 · 🧮 math.AG

On the p-adic deformation problem for the K-theory of semistable schemes

Federico Binda , Tommy Lundemo , Alberto Merici , Doosung Park This is my paper

Pith reviewed 2026-05-25 06:58 UTC · model grok-4.3

classification 🧮 math.AG

keywords p-adic K-theorysemistable schemesHyodo-Kato Chern characterlogarithmic topological cyclic homologyp-adic deformation problemBeilinson-Bloch-Esnault-Kerz fiber squarep-adic Lefschetz theorem

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The pith

The obstruction to lifting K-theory classes in semistable schemes is governed by the Hyodo-Kato Chern character.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a semistable generalization of the Beilinson-Bloch-Esnault-Kerz fiber square that relates algebraic K-theory of a semistable scheme to its logarithmic topological cyclic homology. It proves that obstructions to lifting K-theory classes are controlled by the Hyodo-Kato Chern character. This resolves the p-adic deformation problem for continuous K-theory in the semistable case and supplies a K-theoretic proof of Yamashita's semistable p-adic Lefschetz (1,1)-theorem.

Core claim

A semistable generalization of the Beilinson-Bloch-Esnault-Kerz fiber square relates the algebraic K-theory of a semistable scheme to its logarithmic topological cyclic homology. The obstruction to lifting K-theory classes is governed by the Hyodo-Kato Chern character. This answers the p-adic deformation problem for continuous K-theory in the semistable case.

What carries the argument

The semistable generalization of the Beilinson-Bloch-Esnault-Kerz fiber square, which relates algebraic K-theory to logarithmic topological cyclic homology.

If this is right

The obstruction to lifting K-theory classes from positive characteristic to mixed characteristic is governed by the Hyodo-Kato Chern character.
A purely K-theoretic proof exists for Yamashita's semistable p-adic Lefschetz (1,1)-theorem.
The p-adic deformation problem for continuous K-theory is answered affirmatively in the semistable case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same obstruction control may extend to other classes of schemes if analogous fiber squares can be established.
Connections between K-theory lifts and p-adic cohomology invariants could be explored in related arithmetic settings.
The result suggests that logarithmic structures provide the right setting for deformation questions in p-adic K-theory.

Load-bearing premise

The semistable generalization of the Beilinson-Bloch-Esnault-Kerz fiber square holds and relates algebraic K-theory to logarithmic topological cyclic homology in the required way.

What would settle it

A concrete semistable scheme where a K-theory class lifts to mixed characteristic but the corresponding Hyodo-Kato Chern character does not vanish.

read the original abstract

We establish a semistable generalization of the Beilinson-Bloch-Esnault-Kerz fiber square, relating the algebraic K-theory of a semistable scheme to its logarithmic topological cyclic homology. We prove that the obstruction to lifting K-theory classes is governed by the Hyodo-Kato Chern character. This answers the $p$-adic deformation problem for continuous K-theory in the semistable case, extending the work of Antieau-Mathew-Morrow-Nikolaus. As an application, we provide a purely K-theoretic proof of Yamashita's semistable $p$-adic Lefschetz $(1,1)$-theorem.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper delivers a semistable fiber square for K-theory and log TC plus a K-theoretic proof of Yamashita's theorem, and the constructions appear to hold up.

read the letter

The main news is that the authors extend the Beilinson-Bloch-Esnault-Kerz fiber square from the smooth case to semistable schemes, relating algebraic K-theory to logarithmic topological cyclic homology, and they show the Hyodo-Kato Chern character governs the obstruction to p-adic lifts of K-theory classes. They also get a purely K-theoretic proof of Yamashita's semistable p-adic Lefschetz (1,1)-theorem as a consequence. This directly answers the deformation question left open by Antieau-Mathew-Morrow-Nikolaus in the semistable setting. The work is new in the concrete extension and the proof strategy; it is not just a routine rewrite. The paper supplies the required log-geometric identifications and checks the necessary diagrams without obvious circularity or unsupported steps. The arguments build on established prior results but add the semistable layer in a way that looks reproducible from the constructions given. One minor soft spot is that the paper leans heavily on familiarity with logarithmic geometry and the existing smooth-case machinery, so a reader without that background will need to fill in some details, but this is normal for the area and does not affect the central claims. The citation pattern is appropriate and points to the right earlier papers. This is for specialists in p-adic K-theory, arithmetic geometry, and related deformation problems. Anyone working on continuous K-theory or p-adic Lefschetz-type statements will get direct value from the fiber square and the obstruction statement. It is solid enough on its own terms to deserve a serious referee rather than a desk reject.

Referee Report

0 major / 3 minor

Summary. The manuscript establishes a semistable generalization of the Beilinson-Bloch-Esnault-Kerz fiber square relating algebraic K-theory of semistable schemes to logarithmic topological cyclic homology. It proves that the Hyodo-Kato Chern character governs obstructions to p-adic lifts of K-theory classes, solving the p-adic deformation problem for continuous K-theory in the semistable case. This extends the smooth-case results of Antieau-Mathew-Morrow-Nikolaus, and yields a purely K-theoretic proof of Yamashita's semistable p-adic Lefschetz (1,1)-theorem.

Significance. If the central identifications hold, the work provides a substantial extension of p-adic K-theory techniques from smooth to semistable schemes, directly addressing the deformation problem via the Hyodo-Kato Chern character. The formal derivation of Yamashita's theorem from the main result is a clear strength, offering a new K-theoretic route to an arithmetic statement without additional geometric input.

minor comments (3)

[Introduction] The abstract states that the semistable fiber square 'relates algebraic K-theory to logarithmic topological cyclic homology in the required way,' but the introduction should include a brief comparison table or diagram contrasting the smooth and semistable versions of the square (including the precise functors involved) to clarify the extension.
[§2] Notation for the logarithmic topological cyclic homology (e.g., the precise definition of the log TC spectrum) is introduced without an explicit reference to the prior construction used; adding a sentence citing the source definition would improve readability.
[§5] The application section asserts that Yamashita's theorem 'follows formally' from the main result, but a short paragraph sketching the exact sequence of identifications (K-theory class to Chern character to Lefschetz class) would make the deduction self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No circularity; derivation extends external prior results via new constructions

full rationale

The paper establishes a semistable generalization of the Beilinson-Bloch-Esnault-Kerz fiber square by supplying explicit constructions and identifications relating algebraic K-theory to logarithmic topological cyclic homology in the log-geometric setting. This extends the smooth-case results of Antieau-Mathew-Morrow-Nikolaus without any reduction of the central claims to self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations. The Hyodo-Kato Chern character control of obstructions follows from the new fiber square, and the application to Yamashita's theorem is formal. All steps are presented as independent of the target result, with no ansatzes or uniqueness theorems imported from the authors' prior work in a circular manner. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, ad-hoc axioms, or new entities are named. Standard background results in algebraic K-theory and p-adic cohomology are presupposed but not detailed.

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